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In this paper, we study the distribution of parallelograms and rhombi in agiven set in the plane over arbitrary finite fields $\mathbb{F}_q^2$. As anapplication, we improve a recent result due to Fitzpatrick, Iosevich, McDonald,and Wyman (2021) on the Vapnik-Chervonenkis dimension of the induced distancegraph. Our proofs are based on the discrete Fourier analysis.

Parallelograms and the VC-dimension of the distance sets